[Paper Review] Curvature of almost Hermitian manifolds and applications

Chengjie Yu|arXiv (Cornell University)|Sep 23, 2012

Geometry and complex manifolds13 references3 citations

TL;DR

This paper introduces a local quasi holomorphic frame to derive a curvature formula for almost Hermitian manifolds analogous to that of Hermitian manifolds. Using this formula, it extends results of Wu and Zheng to almost Hermitian geometry, proving that nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds implies the metric splits into a sum of product and holomorphic (2,0)-form components, with a precise codimension formula for the space of such metrics.

ABSTRACT

In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.

Motivation & Objective

• To develop a curvature formula for almost Hermitian manifolds using a new notion of local quasi holomorphic frame.
• To extend known results on holomorphic bisectional curvature from Hermitian to almost Hermitian geometry.
• To classify almost Hermitian metrics with nonpositive holomorphic bisectional curvature on products of compact almost complex manifolds.
• To generalize results of Zheng and the author on curvature-degenerate metrics to the almost complex setting.
• To determine the codimension of the space of such metrics in the product space.

Proposed method

• Introduces a local quasi holomorphic frame adapted to the canonical connection on almost Hermitian manifolds.
• Derives a curvature formula for the canonical connection using this frame, expressing the curvature tensor in terms of second derivatives of the metric and connection terms.
• Applies the curvature formula to prove a subadditivity inequality for curvature forms under metric addition.
• Uses the curvature formula and holomorphic form decomposition to classify metrics with nonpositive holomorphic bisectional curvature on product manifolds.
• Employs a decomposition of holomorphic (2,0)-forms on products into wedge products of holomorphic (1,0)-forms from each factor.
• Establishes a codimension formula by analyzing the space of such metrics relative to the product of individual metric spaces.

Experimental results

Research questions

• RQ1 Can a curvature formula for almost Hermitian manifolds be derived that mirrors the classical formula for Hermitian manifolds?
• RQ2 Does the subadditivity of holomorphic bisectional curvature under metric addition hold in the almost Hermitian setting?
• RQ3 What is the structure of almost Hermitian metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds?
• RQ4 How does the presence of holomorphic (1,0)-forms on the factors constrain the form of such metrics?
• RQ5 What is the real codimension of the space of nonpositively curved metrics on a product manifold relative to the product of individual metric spaces?

Key findings

• A curvature formula for the canonical connection on almost Hermitian manifolds is established using a local quasi holomorphic frame, generalizing the Hermitian case.
• The curvature formula implies that the holomorphic bisectional curvature of a sum of metrics is bounded above by the sum of the individual curvatures.
• Any almost Hermitian metric with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds decomposes as a sum of product metrics and a closed (1,1)-form from holomorphic (2,0)-forms.
• The space of such metrics has real codimension exactly $ 2\dim H^{1,0}(M) \cdot \dim H^{1,0}(N) $ in the product space of metrics on $ M \times N $, when both factors admit nontrivial holomorphic (1,0)-forms.
• If one factor admits no nontrivial holomorphic (1,0)-form, then any such metric must be a product metric.
• The result generalizes Zheng's classification of metrics on complex product manifolds to the almost complex setting.